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A213191 Total sum A(n,k) of k-th powers of parts in all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A213191 #33 May 28 2018 02:58:33
%S A213191 0,0,1,0,1,3,0,1,4,6,0,1,6,9,12,0,1,10,17,20,20,0,1,18,39,44,35,35,0,
%T A213191 1,34,101,122,87,66,54,0,1,66,279,392,287,180,105,86,0,1,130,797,1370,
%U A213191 1119,660,311,176,128,0,1,258,2319,5024,4775,2904,1281,558,270,192
%N A213191 Total sum A(n,k) of k-th powers of parts in all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A213191 In general, if k > 0 then column k is asymptotic to 2^((k-3)/2) * 3^(k/2) * k! * Zeta(k+1) / Pi^(k+1) * exp(Pi*sqrt(2*n/3)) * n^((k-1)/2). - _Vaclav Kotesovec_, May 27 2018
%H A213191 Alois P. Heinz, <a href="/A213191/b213191.txt">Antidiagonals n = 0..140, flattened</a>
%F A213191 A(n,k) = Sum_{j=1..n} A066633(n,j) * j^k.
%e A213191 Square array A(n,k) begins:
%e A213191 :   0,  0,   0,   0,    0,     0,     0, ...
%e A213191 :   1,  1,   1,   1,    1,     1,     1, ...
%e A213191 :   3,  4,   6,  10,   18,    34,    66, ...
%e A213191 :   6,  9,  17,  39,  101,   279,   797, ...
%e A213191 :  12, 20,  44, 122,  392,  1370,  5024, ...
%e A213191 :  20, 35,  87, 287, 1119,  4775, 21447, ...
%e A213191 :  35, 66, 180, 660, 2904, 14196, 73920, ...
%p A213191 b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A213191       add((l->`if`(m=0, l, l+[0, l[1]*p^k*m]))
%p A213191           (b(n-p*m, p-1, k)), m=0..n/p)))
%p A213191     end:
%p A213191 A:= (n, k)-> b(n, n, k)[2]:
%p A213191 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A213191 b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 0, l, l + {0, First[l]*p^k*m}]][b[n - p*m, p - 1, k]], { m, 0, n/p}]]] ; a[n_, k_] := b[n, n, k][[2]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 12 2013, translated from Maple *)
%t A213191 (* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k<n := T[n, k] = T[n-k, k] + PartitionsP[n-k]; T[_, _] = 0; A[n_, k_] := Sum[T[n, j]*j^k, {j, 1, n}]; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 15 2016 *)
%Y A213191 Columns k=0-10 give: A006128, A066186, A066183, A229325, A229326, A229327, A229328, A229329, A229330, A229331, A229332.
%Y A213191 Rows n=0-10 give: A000004, A000012, A052548, A229354, A229355, A229356, A229357, A229358, A229359, A229360, A229361.
%Y A213191 Main diagonal gives A252761.
%Y A213191 Cf. A213180.
%K A213191 nonn,tabl
%O A213191 0,6
%A A213191 _Alois P. Heinz_, Feb 28 2013